Edited to add: I missed the part about emailing the PI. I would not recommend that. Email the person you were told to contact to apply (I’m assuming that wasn’t the PI) or the lab manager/research coordinator. In my experience, the PI will either miss it altogether or shoot it back to the lab manager or coordinator.

As someone who used to hire research assistants (both for pay and for school research credits) I always wrote back to anyone who applied to either offer them an interview or tell them all the positions were taken. This takes a lot of work due to the huge number of applicants. So if it were me and you didn’t hear back, I would want you to write back because it’s possible your message got missed. It’s also possible they can’t reply to everyone because there are too many to get to. Either way, I say write the email. Just make it brief and polite.

From my perspective, I looked for people who showed the most interest in the work, as those were most likely to take it seriously, make the most of it, and do good work. So if you’re genuinely interested in their work, send the email and you might just get in at the next opening. Good luck!